Understanding Geometric Meaning of Ax≤b

[EngWiPy]

Hi,

Suppose we have $$m\times n$$ matrix $$\mathbf{A}$$, and $$n\times 1$$ column vector $$\mathbf{x}$$. Then what do we mean by:

$$\mathbf{A}\mathbf{x}\leq \mathbf{b}$$

geometrically?

Thanks in advance

 

[lurflurf]

It looks like the constraint system if a linear program. The system in most cases defines a polyhedral (that can be shown to be convex) that restricts x to its interior. This occurs in many areas such as sandwich making, in that case we can make n types of sandwiches, we are constained by m conditions on our sandwiches such as our available mustard or number of lemon peal sandwich rolls. We can visualize our posible set of sandwiches by a polyhedra.

 

[EngWiPy]

It looks like the constraint system if a linear program. The system in most cases defines a polyhedral (that can be shown to be convex) that restricts x to its interior. This occurs in many areas such as sandwich making, in that case we can make n types of sandwiches, we are constained by m conditions on our sandwiches such as our available mustard or number of lemon peal sandwich rolls. We can visualize our posible set of sandwiches by a polyhedra.

So, the intersection of all these planes is the polyhedron?

 

[lurflurf]

Yes, in general they can be hyper planes.

 

[EngWiPy]

Yes, in general they can be hyper planes.

Ok thank you

 

[EngWiPy]

Dr. Boyd at Stanford University, says the following in the solution of a homework: The question is: find the maximum volume rectangle $$\mathbf{R}=\{\mathbf{x}:\mathbf{l}\leq\mathbf{x}\leq\mathbf{u}\}$$ in a polyhedron $$\mathbf{P}=\{\mathbf{x}:\mathbf{A}\mathbf{x}\leq\mathbf{b}\}$$.

He says that, an efficient solution would be:

$$\text{max }\left(\prod_{i=1}^n\left(u_i-l_i\right)\right)^{1/n}$$
$$\text{Subject to }\sum_{i=1}^n\left(a_{ij}^+u_j-a_{ij}^-l_j\right)\leq b_i,\,\,\text{ for }i=1,2,\ldots,n$$
where $$a_{ij}^+=\text{max}\{a_{ij},0\}$$ and $$a_{ij}^-=\text{max}\{-a_{ij},0\}$$.

Here I have a couple of questions:

1- I assume that $$u_i-l_i$$ are the dimensions of the rectangle, and so, the volume will be the product of these dimensions, right? Then why we have the power of $$1/n$$ in the objective function?

2- How did he get the constraint? and why?

Any help in these two questions will be highly appreciated.

Thanks